In this section, we present common techniques for the weak stabilization of immersed finite element methods, often referred to as the ghost penalty and generally associated with the Cut Finite Element Method (CutFEM) [1]}, [2]}, [3]}. In the ghost-penalty approach, additional terms are added to the bilinear form which provide control over the solution on the elements that intersect the boundary, while the original approximation space \(V_h\)  is unmodified. These terms are denoted by \(s_h(v_h,w_h)\)  and provide a contribution to the operator norm in the form of \(\Vert  v_h \Vert ^2_{s_h} = s_h(v_h,v_h)\) . The underlying idea is to control the variation of the approximation function across neighboring elements, such that the solution on small cut elements is controlled by the solution on interior elements. A standard way to achieve this is by controlling the jump in the normal derivatives across faces, by adding appropriately scaled stiffnesses to these jumps. Another approach is to add a volumetric penalization to the difference between a function itself and the extension of the function on a neighboring element. In the subsequent sections, both approaches are discussed.
